Large area power transmitter for wireless power transfer

ABSTRACT

A method for wireless power transfer. The method includes adapting a variable form factor transmitter into at least a plurality of cross-coupled segments disposed about a pre-determined wireless power transfer area, wherein the pre-determined wireless power transfer area comprises a dimension exceeding a wavelength corresponding to a characteristic frequency of the variable form factor transmitter, transmitting, from a radio frequency (RF) power source and based at least in part on the characteristic frequency, RF power across the pre-determined wireless power transfer area via a near electromagnetic field of the variable form factor transmitter, and reducing, based on opposing directions of magnetic fields induced by adjacent cross-coupled segments of the plurality of cross-coupled segments, a radiation loss of the wireless power transfer due to a far electromagnetic field of the variable form factor transmitter.

BACKGROUND

Wireless power transfer is the transmission of electrical energy from a power source to an electrical load without the use of man-made conductors to connect the power source to the electrical load. A wireless power transfer system consists of a transmitter and one or more receiver devices. The transmitter is connected to a source of power and converts the power to a time-varying electromagnetic field. The one or more receiver devices receive the power via the electromagnetic field and convert the received power back to an electric current to be utilized by the electrical load.

SUMMARY

In general, in one aspect, the invention relates to a method for wireless power transfer. The method includes adapting a variable form factor transmitter into at least a plurality of cross-coupled segments disposed about a pre-determined wireless power transfer area, wherein the pre-determined wireless power transfer area comprises a dimension exceeding a wavelength corresponding to a characteristic frequency of the variable form factor transmitter, transmitting, from a radio frequency (RF) power source and based at least in part on the characteristic frequency, RF power across the pre-determined wireless power transfer area via a near electromagnetic field of the variable form factor transmitter, and reducing, based on opposing directions of magnetic fields induced by adjacent cross-coupled segments of the plurality of cross-coupled segments, a radiation loss of the wireless power transfer due to a far electromagnetic field of the variable form factor transmitter.

In general, in one aspect, the invention relates to a variable form factor transmitter for wireless power transfer. The variable form factor transmitter includes a plurality of cross-coupled segments disposed about a pre-determined wireless power transfer area, wherein the pre-determined wireless power transfer area comprises a dimension exceeding a wavelength corresponding to a characteristic frequency of the variable form factor transmitter, wherein the plurality of cross-coupled segments are configured to transmit, from a radio frequency (RF) power source and based at least in part on the characteristic frequency, RF power across the pre-determined wireless power transfer area via a near electromagnetic field of the variable form factor transmitter, and reduce, based on opposing directions of magnetic fields induced by adjacent cross-coupled segments of the plurality of cross-coupled segments, a radiation loss of the wireless power transfer due to a far electromagnetic field of the variable form factor transmitter.

In general, in one aspect, the invention relates to a system for wireless power transfer. The system includes a plurality of cross-coupled segments disposed about a pre-determined wireless power transfer area, wherein the pre-determined wireless power transfer area comprises a dimension exceeding a wavelength corresponding to a characteristic frequency of the plurality of cross-coupled segments, and a radio frequency (RF) power source coupled to the plurality of cross-coupled segments, wherein the plurality of cross-coupled segments are configured to transmit, from the RF power source and based at least in part on the characteristic frequency, RF power across the pre-determined wireless power transfer area via a near electromagnetic field of the plurality of cross-coupled segments, and reduce, based on opposing directions of magnetic fields induced by adjacent cross-coupled segments of the plurality of cross-coupled segments, a radiation loss of the wireless power transfer due to a far electromagnetic field of the plurality of cross-coupled segments.

Other aspects of the invention will be apparent from the following transaction description and the appended claims.

BRIEF DESCRIPTION OF DRAWINGS

FIGS. 1A, 1B, and 1C show schematic diagrams of an example system having a variable form factor transmitter in accordance with one or more embodiments of the invention.

FIGS. 2A, 2B, 2C, 2D, 2E, 2F, 2G, 2H, 2J, 2K, 2L, 2M, 2N, and 2P show various diagrams for illustrating an example variable form factor transmitter in accordance with one or more embodiments of the invention.

FIGS. 3A, 3B, 3C, 3D, and 3E show example characteristics of an example variable form factor transmitter in accordance with one or more embodiments of the invention.

FIGS. 4A and 4B show schematic diagrams of example radio frequency (RF) power sources in accordance with one or more embodiments of the invention.

FIGS. 5A, 5B, 5C, 5D, and 5E show schematic and layout diagrams of example receiver devices in accordance with one or more embodiments of the invention.

FIGS. 6A, 6B, 6C, 6D, 6E, 6F, 6G, 7A, 7B, 7C, 7D, 7E, and 7F show schematic and layout diagrams of example power transmitters for wireless power transfer in accordance with one or more embodiments of the invention.

FIG. 8 shows a method flowchart in accordance with one or more embodiments of the invention.

DETAILED DESCRIPTION

Specific embodiments of the invention will now be described in detail with reference to the accompanying figures. Like elements in the various figures are denoted by like reference numerals for consistency.

In the following detailed description of embodiments of the invention, numerous specific details are set forth in order to provide a more thorough understanding of the invention. However, it will be apparent to one of ordinary skill in the art that the invention may be practiced without these specific details. In other instances, well-known features have not been described in detail to avoid unnecessarily complicating the description.

In the following description, any component described with regard to a figure, in various embodiments of the invention, may be equivalent to one or more like-named components described with regard to any other figure. For brevity, at least a portion of these components are implicitly identified based on various legends. Further, descriptions of these components will not be repeated with regard to each figure. Thus, each and every embodiment of the components of each figure is incorporated by reference and assumed to be optionally present within every other figure having one or more like-named components. Additionally, in accordance with various embodiments of the invention, any description of the components of a figure is to be interpreted as an optional embodiment which may be implemented in addition to, in conjunction with, or in place of the embodiments described with regard to a corresponding like-named component in any other figure. In the figures, black solid collinear dots indicate that additional components similar to the components before and/or after the solid collinear dots may optionally exist.

Throughout the application, ordinal numbers (e.g., first, second, third, etc.) may be used as an adjective for an element (i.e., any noun in the application). The use of ordinal numbers is not to imply or create any particular ordering of the elements nor to limit any element to being only a single element unless expressly disclosed, such as by the use of the terms “before”, “after”, “single”, and other such terminology. Rather, the use of ordinal numbers is to distinguish between the elements. By way of an example, a first element is distinct from a second element, and the first element may encompass more than one element and succeed (or precede) the second element in an ordering of elements.

In general, embodiments of the invention provide a method, a transmitter device, and a system for wireless power transfer. In one or more embodiments of the invention, the method, transmitter device, and system is based on a power transmitter that includes a number of capacitors and inductive segments disposed along a path that defines a wireless power transfer area. The capacitors are connected in series via at least the inductive segments into a string of distributed capacitors. In one or more embodiments of the invention, the string of distributed capacitors is integrated onto a laminated material sheet (e.g., including at least dielectric material) encompassing at least the path that defines the power transfer area. In one or more embodiments, one or more capacitors and one or more inductive segments are constructed using conductive strips attached to two opposing surfaces of the material sheet. For example, a capacitor may include two conductive strips attached to the two opposing surfaces. Further, an inductive segment may include another conductive strip attached to one of the two opposing surfaces. Accordingly, RF power is transmitted, from a radio frequency (RF) power source and based at least on a characteristic frequency of the string of distributed capacitors, across the wireless power transfer area via a near electromagnetic field of the of the string of distributed capacitors. In one or more embodiments of the invention, the characteristic frequency is within the industrial, scientific and medical (ISM) radio band defined by the International Telecommunication Union (ITU) Radio Regulations. For example, the characteristic frequency may be within the type A frequency range (i.e., 6.765 MHz-6.795 MHz) defined in the ITU Radio Regulations Article 5, footnote 5.138.

In one or more embodiments of the invention, the power transmitter has a fixed form factor specific to a pre-determined wireless power transfer area. In one or more embodiments of the invention, the power transmitter is adaptable into different form factors (referred to as adapted form factors) to fit different wireless power transfer areas. In such embodiments, the power transmitter is a variable form factor transmitter with a characteristic frequency that is maintained to be substantially independent of the adapted form factors. For example, the characteristic frequency may be maintained, as the adapted form factor varies, within the ISM radio band.

FIG. 1A shows a schematic diagram of an example system (100) in accordance with one or more embodiments of the invention. In one or more embodiments, one or more of the modules and elements shown in FIG. 1A may be omitted, repeated, and/or substituted. Accordingly, embodiments of the invention should not be considered limited to the specific arrangements of modules shown in FIG. 1A.

As shown in FIG. 1A, the system (100) includes a variable form factor transmitter (102) receiving power from an RF power source (108) for wireless power transfer across a wireless power transfer area (101) having one or more receiver devices (e.g., denoted as circular icons labeled as A, B, C, D, E, and F) disposed therein. Each of these components is described in detail below.

In one or more embodiments of the invention, the wireless power transfer area (101) is any three dimensional (3D) physical space where the one or more receiver devices are receiving power from the variable form factor transmitter (102). For example, the wireless power transfer area (101) may include a 3D space within a building or a vehicle, such as a room, a hallway, a passenger cabin of a car, bus, train, airplane, or space ship, or any portion of the building or vehicle. In another example, the wireless power transfer area (101) may include a 3D space that is not enclosed, such as a play ground, a roadway, an amusement park, or any type of field on the ground, above the ground, or away from the earth in the space (e.g., an atmospheric layer or interstellar space). In yet another example, the wireless power transfer area (101) may include an underground or under-water space, such as a cave, an under water region near an ocean platform or sea bed, etc. In still another example, the wireless power transfer area (101) may include a combination of the examples above.

In one or more embodiments of the invention, the variable form factor transmitter (102) is disposed entirely within the wireless power transfer area (101), overlaps the wireless power transfer area (101), or in the vicinity of the wireless power transfer area (101). In one or more embodiments, at least a portion of the variable form factor transmitter (102) may be inserted in a protective sleeve, embedded in a material sheet, free-standing in the wireless power transfer area (101), or attached to the wireless power transfer area (101). In one or more embodiments, at least a portion of the variable form factor transmitter (102) may be stationery or moving with respect to the wireless power transfer area (101) and/or the one or more receiver devices (e.g., denoted as circular icons labeled as A, B, C, D, E, and F) disposed therein. In one or more embodiments of the invention, the form factor of the variable form factor transmitter (102) is adapted according to a geometric constraint imposed by the wireless power transfer area (101). For example, the variable form factor transmitter (102) may be made of pliable material such that the form factor of the variable form factor transmitter (102) is changed by the user to fit the physical shape of the room, hallway, passenger cabin, play ground, roadway, amusement park, field, cave, under-water region, etc. of the wireless power transfer area (101). In this context, the form factor of the variable form factor transmitter (102) is based on the wireless power transfer area (101). For example, the form factor of the variable form factor transmitter (102) may include a 3D portion, such as a curved surface, a helical curve, etc.

In one or more embodiments of the invention, the receiver devices (A) through (F) may be of the same type or of different types that are used by one or more users, such as individual persons. In one or more embodiments, one or more of the receiver devices (A) through (F) are disposed at user specified locations throughout the wireless power transfer area (101) and are stationary during the wireless power transfer. In one or more embodiments, one or more of the receiver devices (A) through (F) have dimensions that are smaller than the dimensions of the wireless power transfer area (101). In one or more embodiments, one or more of the receiver devices (A) through (F) have dimensions that are comparable to or greater than the dimensions of the wireless power transfer area (101). For example, the receiver device (A) may be a lighting device placed on the ceiling of a room or hallway by the user. In one or more embodiments, one or more of the receiver devices (A) through (F) are carried by respective users who move around throughout the wireless power transfer area (101) from time to time during the wireless power transfer. Based on the nature of the near electromagnetic field of the variable form factor transmitter (102), the power of the near electromagnetic field that is not received by any of the receiver device is returned to the variable form factor transmitter (102) and the RF power source (108). This is in contrast to a far electromagnetic field via which power is radiated resulting in energy loss that is not productive for the wireless power transfer. Examples of the receiver device (A), receiver device (B), receiver device (C), receiver device (D), receiver device (E), and receiver device (F) are described in reference to FIGS. 5A, 5B, 5C, 5D, and 5E below.

In one or more embodiments of the invention, the variable form factor transmitter (102) includes a string of distributed capacitors. In particular, the string of distributed capacitors includes multiple capacitor-wire segments that are connected in series to conduct radio-frequency (RF) electrical current (105) generated by the power source (108). The RF electrical current (105) induces magnetic fields (e.g., magnetic field (106)) that are present throughout the wireless power transfer area (101). In one or more embodiments, the string of distributed capacitors is disposed along a path such that the magnetic fields throughout the wireless power transfer area (101) exceeds a threshold that is based on a power requirement of the receiver devices. In this context, the path is based on the wireless power transfer area (101). In one or more embodiments, the RF electrical current (105) enters/exits the wire at a terminal A (204 a) and a terminal B (204 b). In one or more embodiments, additional intervening components (not shown) may also be inserted in the series of capacitor-wire segments or inserted between the series of capacitor-wire segments and one or more terminals (e.g., terminal A (204 a), terminal B (204 b)) without impeding the operation of the variable form factor transmitter (102).

In one or more embodiments, each capacitor-wire segment includes a capacitor (e.g., capacitor (103)) connected to a wire segment (e.g., wire segment (104)). In one or more embodiments, each capacitor (e.g., capacitor (103)) in the variable form factor transmitter (102) has the same nominal capacitance value, as any other capacitor therein, that is determined prior to disposing the variable form factor transmitter (102) in the wireless power transfer area (101). For example, the capacitors (e.g., capacitor (103)) in the variable form factor transmitter (102) may be installed in a factory before a user uses the variable form factor transmitter (102) to provide power wirelessly within the wireless power transfer area (101). The capacitors (e.g., capacitor (103)) may be of a suitable type, such as ceramic capacitors, film and paper capacitors, electrolyte capacitors, polymer capacitors, silver mica capacitors, etc. In one or more embodiments, one or more of the capacitors may include two aluminum or other metallic sheets, foils, or films separated by an aluminum or other metallic oxide layer. As is typical in a factory manufacturing process, the capacitance values of all capacitors (e.g., capacitor (103)) in the variable form factor transmitter (102) may vary within a range (referred to as a capacitance range), e.g., due to a manufacturing tolerance.

In one or more embodiments, each capacitor-wire segment includes a wire segment having a pre-determined segment length and a pre-determined inductance per unit length. For example, the wire segments (e.g., wire segment (104)) in the variable form factor transmitter (102) may be installed in a factory before a user uses the variable form factor transmitter (102) to provide power wirelessly within the wireless power transfer area (101). The wire segments (e.g., wire segment (104)) may be of a suitable type, such as insulated or un-insulated wires, sheets, foil, or films made of copper, aluminum, or other suitable metal and/or alloy material. In one or more embodiments, one or more of the wire segments (e.g., wire segment (104)) are flexible or pliable such that the user may bend, stretch, or otherwise change the shape of the one or more wire segments. As is typical in a factory manufacturing process, the length and inductance values of each and all wire segments (e.g., wire segment (104)) in the variable form factor transmitter (102) may vary within a range (referred to as a length range and an inductance range), e.g., due to a manufacturing tolerance.

In one or more embodiments of the invention, by confining the electrical fields, the capacitors (e.g., capacitor (103)) in the variable form factor transmitter (102) reduce stray electric fields and the resultant induced voltage of the wire segments (e.g., wire segment (104)). Accordingly, the capacitors (e.g., capacitor (103)) in the variable form factor transmitter (102) reduce the fraction of energy stored in the stray capacitance of the wire segments (e.g., wire segment (104)) over the total energy in the system (100). The reduction of both induced voltage and stored energy associated with the stray capacitance reduces loss due to environmental interactions and improves safety for the user.

In one or more embodiments of the invention, the variable form factor transmitter (102) is associated with a characteristic frequency that is based at least on the pre-determined capacitance, the pre-determined segment length, and the pre-determined inductance per unit length. The characteristic frequency of the variable form factor transmitter (102) is described in reference to FIGS. 2A, 2B, 2D, 2E, 3A, 3B, 3C, 3D, and 3E below. Throughout this document, the terms “characteristic frequency” and “resonant frequency” may be used interchangeably depending on context.

In one or more embodiments, instead of the direct connection to the power source (108), the variable form factor transmitter (102) receives power from the power source (108) using inductive coupling via a driving loop (109 a). FIG. 1B shows a schematic diagram of the example system (100) in the inductive coupling power configuration. Details of receiving power via the driving loop (109 a) are described in reference to FIG. 1C below.

FIG. 1C shows a schematic diagram of supplying power via the driving loop (109 a) depicted in FIG. 1B above. In one or more embodiments, one or more of the modules and elements shown in FIG. 1C may be omitted, repeated, and/or substituted. Accordingly, embodiments of the invention should not be considered limited to the specific arrangements of modules shown in FIG. 1C.

As shown in FIG. 1C, the driving loop (109 a) includes one or more loops of conducting wire (e.g., having an inductance L₁) that are coupled to the power source (108) via a balun (108 a). The balun (108 a) includes a tuning capacitor A (109 d) (e.g., having a variable capacitance C₁), a tuning capacitor B (109 e) (e.g., having a variable capacitance C₂), and a coaxial cable (109 c) (e.g., coiled around a ferrite core (109 b) and having an inductance L₂). Specifically, the driving loop (109 a) is placed at a distance (110) from the variable form factor transmitter (102) such that the power source (108) supplies power to the variable form factor transmitter (102) via electromagnetic coupling across the distance (110). In one or more embodiments, the tuning capacitor B (109 e) is tuned to resonate with inductance L₂ of the ferrite core (109 b) to form a parallel resonant LC circuit, which imposes a high impedance between the two opposite ends of the coaxial cable (109 c). Further, the tuning capacitor A (109 d) is used to tune the resonant frequency of the driving loop (109 a) to match the frequency of the RF power source (108). The distance (110) between the driving loop (109 a) and the variable form factor transmitter (102) may be adjusted in order to match the apparent input impedance of variable form factor transmitter (102) to the impedance of the coaxial cable (109 c), and the output impedance of the RF power source (108).

FIG. 2A shows a schematic diagram of a parallel-wire transmission line (201) in accordance with one or more embodiments of the invention. In one or more embodiments, one or more of the modules and elements shown in FIG. 2A may be omitted, repeated, and/or substituted. Accordingly, embodiments of the invention should not be considered limited to the specific arrangements of modules shown in FIG. 2A.

As shown in FIG. 2A, the sinusoid-shaped icons (201 a) and (201 b) represent electromagnetic waves propagating along the parallel-wire transmission line (201). The parallel-wire transmission line (201) is composed of two parallel wires (201 d) each having wire segments joined by capacitors, where s denotes the length of each wire segment, C denotes the capacitance of each capacitor, and q denotes the electric charge displacement along the parallel-wire transmission line (201). In the context that the two parallel wires (201 d) conduct RF current (e.g., electrical current (105) depicted in FIG. 1A), each wire of two parallel wires (201 d) is also referred to as a conductor wire throughout this document. The distance between the sinusoid-shaped icons (201 a) and (201 b) corresponds to the length of the parallel-wire transmission line (201) while the spacing between the two parallel string of capacitors corresponds to the width of the parallel-wire transmission line (201). While the length of the parallel-wire transmission line (201) may be comparable to a length of other dimension of the wireless power transfer area (101), the width of the parallel-wire transmission line (201) may range from less than one centimeter to a width or other dimension of the wireless power transfer area (101). In one or more embodiments, the parallel-wire transmission line (201) corresponds to a portion of the variable form factor transmitter (102) depicted in FIG. 1A above. In other words, two sections of the string of distributed capacitor depicted in FIG. 1A may be disposed parallel to each other. Generally, the electric charge, q, displaced along the parallel-wire transmission line (201) is a function of a position along the parallel-wire transmission line (201) and time. The corresponding charge density (i.e., electric charge per unit length), ρ_(λ), and electrical current, I, are given by Eq. (1) below for the parallel-wire transmission line (201). In Eq. (1), x and t denote the position along the parallel-wire transmission line (201) and time, respectively.

$\begin{matrix} {{\rho_{\lambda} = {q^{\prime} = \frac{\partial q}{\partial x}}},{I = {\overset{.}{q} = \frac{\partial q}{\partial t}}}} & {{Eq}.\mspace{20mu}(1)} \end{matrix}$

TABLE 1 shows additional definitions of variables used in the equations throughout this document.

TABLE 1   c = capacitance per milt length l = inductance per unit length C = capacitance of each joining capacitor s = length of each segment q = charge displacement ρ_(λ) = charge density λ = wavelength in free space I = current U_(j) = energy stored in the two joining capacitors u_(E) = electrical energy stored per unit length u_(B) = magnetic energy stored per unit length v = asymptotic velocity ω₀ = cutoff frequency v_(p) = phase velocity v_(g) = group veloctiy

The electrical energy, U_(j), stored in a pair of adjoining capacitors (e.g., capacitor pair (201 c)) in the parallel-wire transmission line (201) is given by Eq. (2) below.

$\begin{matrix} {U_{j} = {{{2 \cdot \frac{1}{2}}\frac{q^{2}}{C}} = \frac{q^{2}}{C}}} & {{Eq}.\mspace{14mu}(2)} \end{matrix}$

In the scenario where s is substantially less than the spatial variation of q, the stored energy, U_(j), divided by the segment length, s, may be considered as a density of energy stored in the capacitors, C, along the parallel-wire transmission line (201). Let c denote the stray capacitance per unit length between the two parallel wires of the parallel-wire transmission line (201). The total electrical energy, u_(E), stored per unit length along the parallel-wire transmission line (201) is given by Eq. (3) below.

$\begin{matrix} {U_{E} = {{\frac{1}{2}\frac{\rho_{\lambda}^{2}}{C}} = \frac{q^{2}}{sC}}} & {{Eq}.\mspace{14mu}(3)} \end{matrix}$

The total magnetic energy, u_(B), stored per unit length along the parallel-wire transmission line (201) is given by Eq. (4) below. u _(B)=½lI ²  Eq. (4)

Accordingly, the Lagrangian of the parallel-wire transmission line (201) is given by Eq. (5) below.

$\begin{matrix} {\begin{matrix} {\mathcal{L} = {{U_{E} - U_{B}} = {\int{{dx}\left( {\mathcal{U}_{E} - \mathcal{U}_{B}} \right)}}}} \\ {= {\int{{dx}\left\lbrack {{\frac{1}{2}\frac{\rho_{\lambda}^{2}}{C}} + \frac{q^{2}}{sC} - {\frac{1}{2}{lI}^{2}}} \right\rbrack}}} \\ {= {\int{{dx}\left\lbrack {{\frac{- 1}{2}q\frac{q^{''}}{C}} + \frac{q^{2}}{sC} - {\frac{1}{2}l{\overset{.}{q}}^{2}}} \right\rbrack}}} \end{matrix}\quad} & {{Eq}.\mspace{14mu}(5)} \end{matrix}$

The generalized momentum π, the Euler-Lagrange equation of motion, and the wave equation of the parallel-wire transmission line (201) are given by Eq. (6), Eq. (7), and Eq. (8) below.

$\begin{matrix} {\pi = {{\partial_{\overset{.}{q}}} = {{- l}\overset{.}{q}}}} & {{Eq}.\mspace{14mu}(6)} \\ {\overset{.}{\pi} = {{\partial_{q}} = {{{- l}\overset{¨}{q}} = {{- \frac{q^{''}}{c}} + {2\frac{q}{sC}}}}}} & {{Eq}.\mspace{14mu}(7)} \\ {{- \overset{¨}{q}} = {{- \frac{q^{''}}{lc}} + {2\frac{q}{lsC}}}} & {{Eq}.\mspace{14mu}(8)} \end{matrix}$

Based on the wave equation Eq. (8), the dispersion relation for the parallel-wire transmission line (201) is given by Eq. (9a), Eq. (9b), and Eq. (9c) below.

$\begin{matrix} {v \equiv \frac{1}{\sqrt{lc}}} & {{Eq}.\mspace{14mu}\left( {9a} \right)} \\ {\omega_{0} = \frac{1}{\sqrt{{lsC}/2}}} & {{Eq}.\mspace{14mu}\left( {9b} \right)} \\ {\omega^{2} = {{v^{2}k^{2}} + \omega_{0}^{2}}} & {{Eq}.\mspace{14mu}\left( {9c} \right)} \end{matrix}$

In Eq. (9a), Eq. (9b), and Eq. (9c), ω represents an angular frequency, k represents a wave number, v represents an asymptotic wave velocity as defined in Eq. (9a), and ω_(o) represents a cut off angular frequency as defined in Eq. (9b). In particular, the cut off angular frequency ω_(o) is independent of the length, and varies logarithmically with the width, of the parallel-wire transmission line (201). In one or more embodiments, one or more wire segments with associated capacitors of the parallel-wire transmission line (201) are detachable. Accordingly, the parallel-wire transmission line (201) may be reconfigured, without substantially changing ω_(o), by the user to change the total length according to the dimension of the wireless power transfer area (101).

Based on Eq. (9c), FIG. 3A shows a plot of angular frequency, ω, versus wave number, k, to illustrate the dispersion relation for the parallel-wire transmission line (201). In addition, the phase velocity, v_(p), and group velocity, v_(g), are given in Eq. (10a) and Eq. (10b) below.

$\begin{matrix} {v_{p} = \frac{\omega}{k}} & {{Eq}.\mspace{14mu}\left( {10a} \right)} \\ {v_{g} = \frac{\partial\omega}{\partial k}} & {{Eq}.\mspace{14mu}\left( {10b} \right)} \end{matrix}$

Note that as the wave number k asymptotically approaches 0, the phase velocity v_(p) asymptotically approaches infinity, the group velocity v_(g) asymptotically approaches 0, and the angular frequency co asymptotically approaches ω₀.

FIG. 2B shows a schematic diagram of the parallel-wire transmission line (201) driven by the RF power source (108) in accordance with one or more embodiments of the invention. In one or more embodiments, one or more of the modules and elements shown in FIG. 2B may be omitted, repeated, and/or substituted. Accordingly, embodiments of the invention should not be considered limited to the specific arrangements of modules shown in FIG. 2B.

As shown in FIG. 2B, the parallel-wire transmission line (201) is driven by the RF power source (108) connected via the terminal A (204 a) and terminal B (204 b). Further, the parallel-wire transmission line (201) is terminated by an electrically conducting connection (202) and operating at the characteristic frequency ω_(o). In one or more embodiments of the invention, the electrically conducting connection (202) may be substituted by a variable capacitor or other electronic component, which may be used to fine tune the characteristic frequency of the parallel-wire transmission line (201).

In one or more embodiments of the invention, the configuration of the parallel-wire transmission line (201) shown in FIG. 2B approximates the variable form factor transmitter (102) depicted in FIG. 1A above. Similar to FIG. 1A, receiver devices (e.g., denoted as circular icons labeled as A, B, C, D, E, and F) are disposed about the parallel-wire transmission line (201) shown in FIG. 2B. The approximation is particularly suitable for the scenario where the wireless power transfer area (101) has an elongated shape and where the string of distributed capacitors of the variable form factor transmitter (102) is arranged into a pair of parallel lines according to the elongated shape of the wireless power transfer area (101). As described below, the characteristic frequency of the variable form factor transmitter (102) corresponds to ω_(o) described in reference to FIG. 2A above and is substantially independent of the length, and varies logarithmically with the width, of the parallel-wire transmission line (201).

In the configuration shown in FIG. 2B, the standing wave along the parallel-wire transmission line (201), as excited by the RF power source (108), has an infinite phase velocity. Therefore, the voltages and currents along the parallel-wire transmission line (201) are all in phase at different positions of the parallel-wire transmission line (201). In other words, the effective electrical length of the parallel-wire transmission line (201) equals zero regardless of the physical length of the parallel-wire transmission line (201). In the scenario where there is no energy loss in the parallel-wire transmission line (201), the input impedance of the parallel-wire transmission line (201) as presented to the RF power source (108) equals zero regardless of the physical length of the parallel-wire transmission line (201). In other words, the parallel-wire transmission line (201) is equivalent to an RLC circuit (not shown) resonant at ω_(o), regardless whether the physical length of the parallel-wire transmission line (201) is much shorter or much longer than the free-space wavelength (e.g., based on the transmission medium of the wireless power transfer area (101)) of the driving frequency, i.e., ω_(o). Accordingly, the parallel-wire transmission line (201) driven by the RF power source (108) and terminated by the electrically conducting connection (202) may be used as a resonant power source for wireless power transfer to induce resonances of receiver devices that are placed in the vicinity of the parallel-wire transmission line (201). In particular, the resonant receiver devices couple to the electric and/or magnetic fields generated by the standing wave of the parallel-wire transmission line (201) and receive power from the electric and/or magnetic fields.

In one or more embodiments, the resonant receiver devices receive power from a near electromagnetic field of the parallel-wire transmission line (201). Even if the physical length of the parallel-wire transmission line (201) is much longer than the free-space wavelength (e.g., based on the transmission medium of the wireless power transfer area (101)) of the driving frequency, the power supplied from the RF power source (108) is substantially retained in the parallel-wire transmission line (201) for transferring to the nearby resonant receiver devices without being lost to far field radiation. The quality factor of the parallel-wire transmission line due to radiation loss depends only on the wire separation and wire radius, not on the length.

FIG. 2C shows a variation of the parallel-wire transmission line (201) with distributed capacitance in which one of the conductor wires forms a conducting shield (203) that surrounds the other conductor wire, hereafter referred to as the shielded transmission line (201 a). For example, the conducting shield (203) may be substantially cylindrical. The shielded transmission line (201 a) shown in FIG. 2C operates by the same principle as the parallel-wire transmission line (201) shown in FIG. 2B above, except the distributed capacitance is only placed on the center conductor. In some configurations, the center conductor may not be concentric with the outer conductor (i.e., conducting shield 203). Further, the cross sections of the center conductor and outer conductor (i.e., conducting shield 203) may not be circular.

In one or more embodiments of the invention, the configuration of the shielded transmission line (201 a) shown in FIG. 2C approximates the variable form factor transmitter (102) depicted in FIG. 1A above. Similar to FIG. 1A, receiver devices (e.g., denoted as circular icons labeled as A, B, C, D, E, and F) are disposed about the parallel-wire transmission line (201) shown in FIG. 2C. The approximation is particularly suitable for the scenario where the wireless power transfer area (101) corresponds to the interior space within a conductive enclosure, such as within a metal pipeline, an airframe of an airplane or space shuttle, etc. The characteristic frequency of the variable form factor transmitter (102), as shown in FIG. 2C, corresponds to ω_(o) described in reference to FIGS. 2A and 2B above and is substantially independent of the length, and varies logarithmically with the diameter, of the conducting shield (203). The characteristic frequency of the shielded transmission line (201 a) shown in FIG. 2C is given by Eq. (11). Note that this differs from Eq. (9b) by a factor of √{square root over (2)} due to the fact that only one of the conductor wires includes distributed capacitors.

$\begin{matrix} {\omega_{0} = \frac{1}{\sqrt{lsC}}} & {{Eq}.\mspace{14mu}(11)} \end{matrix}$

FIG. 3B shows a plot of the quality factor, Q, of a parallel-wire transmission line (e.g., shown in FIG. 2A or FIG. 2B) of arbitrary length, consisting of 14 AWG copper wire, driven at 6.78 MHz, as a function of the separation, d, (between the two wires) divided by the free-space wavelength λ. For wire separations large relative to the free-space wavelength, the Q is suppressed due to radiation loss. However, for wire separations small compared to the free-space wavelength, the radiation is suppressed and the loss is dominated by ohmic losses in the copper wire.

Note that the shielded transmission line (201) has no radiative loss due to the fact that the conducting shield (203) completely encloses the internal electromagnetic fields.

In contrast, while a conducting wire loop driven by the RF power source (108), described in reference to FIG. 2D below, may also transfer power to resonant receiver devices in the vicinity, the efficiency of the power transfer is decreased due to far field radiation as the dimension of the conducting wire loop increases to approach or exceed the free-space wavelength of the driving frequency. FIG. 3C shows a plot of the quality factor, Q, of a circular loop consisting of 14 AWG copper wire, driven at 6.78 MHz, as a function of the loop radius a divided by the free-space wavelength λ. Note that the Q becomes low, and therefore the efficiency of wireless power transfer is suppressed, as the loop radius becomes large relative to the free-space wavelength.

FIG. 2D shows a schematic diagram of a wire loop (204) having distributed capacitors and driven by the RF power source (108) in accordance with one or more embodiments of the invention. In one or more embodiments, one or more of the modules and elements shown in FIG. 2D may be omitted, repeated, and/or substituted. Accordingly, embodiments of the invention should not be considered limited to the specific arrangements of modules shown in FIG. 2D.

In one or more embodiments, the wire loop (204) has a circular loop radius, a, and a wire radius (corresponding to a gauge of the wire), b, (not shown) and is composed of wire segments of length s joined by a number of capacitors, C. In one or more embodiments of the invention, the configuration of the wire loop (204) shown in FIG. 2D approximates the variable form factor transmitter (102) depicted in FIG. 1A above. The approximation is particularly suitable for the scenario where a particular shape of the wireless power transfer area (101) matches the circular form factor of the variable form factor transmitter (102). As is described below, the characteristic frequency of the variable form factor transmitter (102) corresponds to a resonant frequency ω_(o) of the wire loop (204) and is substantially independent of the width and/or length (i.e., form factor) of the wire loop (204).

The inductance, L, the total capacitance, C_(tot), and the resonant angular frequency, ω_(o), of the wire loop (204) are given by Eq. (12a), Eq. (12b), and Eq. (12c) below.

$\begin{matrix} {L = {\mu\;{a\left\lbrack {{\ln\left( \frac{8a}{b} \right)} - 2} \right\rbrack}}} & {{Eq}.\mspace{14mu}\left( {12a} \right)} \\ {C_{tot} = {\frac{C}{N} = \frac{C}{\left( {2\pi\;{a/s}} \right)}}} & {{Eq}.\mspace{14mu}\left( {12b} \right)} \\ \begin{matrix} {\omega_{0}^{2} = {\frac{1}{{LC}_{tot}} = \frac{1}{\mu\;{{a\left\lbrack {{\ln\left( \frac{8a}{b} \right)} - 2} \right\rbrack} \cdot \frac{s}{2\;\pi\; a} \cdot C}}}} \\ {= \frac{1}{\frac{\mu}{2\;\pi}{{sC}\left\lbrack {{\ln\left( \frac{8a}{b} \right)} - 2} \right\rbrack}}} \end{matrix} & {{Eq}.\mspace{14mu}\left( {12c} \right)} \end{matrix}$

In Eq. (12a), Eq. (12b), and Eq. (12c), N denotes the number of wire segments or capacitors, C, in the wire loop (204) and μ denotes the electro-magnetic permeability of the transmission medium in the wireless power transfer area (101). In one or more embodiments, the resonant angular frequency, ω_(o), depends only weakly on the radius, a, of the wire loop (204) or the wire radius, b. In one or more embodiments, one or more wire segments with associated capacitors of the wire loop (204) are detachable. Accordingly, the wire loop (204) may be reconfigured, without substantially changing the resonant angular frequency ω_(o), by the user to change the loop radius, a, according to the dimensions of the wireless power transfer area (101).

Unlike the parallel-wire transmission line (201) shown in FIG. 2A above, the wire loop (204) becomes an efficient far field radiator as the radius, a, becomes comparable to or exceeds the free-space wavelength (e.g., based on the transmission medium of the wireless power transfer area (101)) of the driving frequency, i.e., ω_(o). The radiation resistance (i.e., effective series resistance due to far field radiation) R_(rad) of a closed loop of wire carrying a uniform current is given by the double integral over the wire path shown Eq. (13a) below.

$\begin{matrix} {R_{rad} = {\frac{{\zeta\kappa}^{2}}{4\;\pi}{\int{{{dr}_{1} \cdot {dr}_{2}}\frac{\sin\left( {\kappa{{r_{1} - r_{2}}}} \right)}{\kappa{{r_{1} - r_{2}}}}}}}} & {{Eq}.\mspace{14mu}\left( {13a} \right)} \\ {\zeta = \sqrt{\frac{\mu}{\epsilon}}} & {{Eq}.\mspace{14mu}\left( {13b} \right)} \\ {{\kappa \equiv \frac{\omega}{c}} = \frac{2\pi}{\lambda}} & {{Eq}.\mspace{14mu}\left( {13c} \right)} \end{matrix}$

In Eq. (13a), based on the transmission medium of the wireless power transfer area (101), ζ is the impedance of free space, and κ is the free-space wavenumber. Based on Eq. (13a) applied to the wire loop (204), FIG. 3D shows a plot of radiation resistance divided by the impedance of free space as a function of radius divided by wavelength. As can be seen from FIG. 3D, the radiation resistance has the asymptotic forms for large and small loop radius given in Eq. (14) below.

$\begin{matrix} {{{\frac{R_{rad}}{\zeta} \approx {\frac{8\;\pi^{5}}{3}\left( \frac{a}{\lambda} \right)^{4}}},{a ⪡ \lambda}}{{\frac{R_{rad}}{\zeta} \approx {\pi^{2}\left( \frac{a}{\lambda} \right)}},{a ⪢ \lambda}}} & {{Eq}.\mspace{14mu}(14)} \end{matrix}$

The quality factor, Q, of the loop due to radiation is equal to the ratio of the inductive reactance, ω_(o)L, divided by the total series resistance, R, which includes the radiation resistance, R_(rad). As the radiation resistance increases, the quality factor decreases, causing the efficiency of the wireless power transfer to decrease.

For the circular wire loop (204) shown in FIG. 2D, Eq. (12c) applies where ω₀=√{square root over (2π/(μsC(ln(8a/b)−2)))} with a being the loop radius and b being the wire radius. For the parallel-wire transmission line (201) shown in FIG. 2B, Eq. (9b) applies and it can be shown that ω₀=√{square root over (2π/(μsCln(d/b)−2)))}, with d being the width of the parallel-wire transmission line and b being the wire radius. The characteristic frequencies, ω₀, have similar values for both circular loop and parallel-wire configurations if ln(a/b)≈ln(d/b). In this manner, a single variable form factor transmitter (102) may be manufactured for use in both elongated-shaped service area and circular-shaped service area based on the user adapted elongated form factor or circular form factor. In other words, based on the wire diameter, b, used to manufacture the variable form factor transmitter (102), the user may select the loop radius, a, and the parallel-wire transmission line width, d, such that ln(a/b)≈ln(d/b). In this manner, one single variable form factor transmitter manufactured in the factory can be configured into either a parallel-wire form factor depicted in FIG. 2B or a circular form factor depicted in FIG. 2D to supply power to the same set of receiving devices that are tuned to the particular resonant frequency, ω₀.

FIG. 2E shows a schematic diagram of a rectangular loop (206) having distributed capacitors and driven by the RF power source (108) in accordance with one or more embodiments of the invention. In one or more embodiments of the invention, the configuration of the rectangular loop (206) approximates the variable form factor transmitter (102) depicted in FIG. 1A above. Similar to FIG. 1A, receiver devices (e.g., denoted as circular icons labeled as A, B, C, D, E, and F) are disposed about the rectangular loop (206) shown in FIG. 2E. For example, the rectangular loop (206) may correspond to the parallel-wire transmission line (201) shown in FIG. 2B that has been adapted by a user to fit a rectangular-shaped wireless power transfer area. In another example, the rectangular loop (206) may correspond to the wire loop (204) shown in FIG. 2D that has been adapted by a user to fit a rectangular-shaped wireless power transfer area. As shown in FIG. 2E, the rectangular loop (206) is driven by the RF power source (108) using a transformer coupling scheme. In particular, the transformer (206 a) includes a capacitor C₂ in parallel to the primary coil L₁ and a capacitor C₁ in parallel to the secondary coil L₁. In addition, the electrically conducting connection (202) shown in FIG. 2B is substituted by a capacitor C₂. The capacitance values of the capacitors C₁, C₂, and C₃ may be adjusted in the factory and/or by the user for impedance matching between the power source (108) and the rectangular loop (206) and for tuning the resonant frequency of the rectangular loop (206).

FIG. 2F shows a schematic diagram of connecting the power source (108) using a capacitive coupling scheme. In particular, the power source (108) is connected to a distributed capacitor string (207), via a coaxial cable (208) and a twisted pair (209), at opposite terminals of a tuning capacitor C₁. The value of the tuning capacitor C₁ may be adjusted in the factory or by the user to provide a proper impedance match to both the RF power source (108) and the coaxial cable (208). By attaching the shield of the coaxial cable (208) to a voltage node of the distributed-capacitor string (207), the shield of the coaxial cable (208) is maintained at ground potential.

In one or more embodiments, the distributed capacitor string (207) may correspond to a portion of the parallel-wire transmission line (201) shown in FIGS. 2B and 2C, a portion of the wire loop (204) shown in FIG. 2D, or a portion of the rectangular loop (206) shown in FIG. 2E. The voltage magnitude relative to ground (210) induced by the power source (108), is shown as a function of the position along the distributed capacitor string (207).

FIG. 2G shows a schematic diagram for connecting the power source (108) to the variable form factor transmitter using an alternative capacitive coupling scheme. As shown in FIG. 2G, a resonant balun (211) is used to connect the power source (108) to the tuning capacitor, C₁.

FIG. 3E is a plot of inductance as a function of the aspect ratio (represented by width/half-perimeter) of a rectangular loop (e.g., rectangular loop (206) depicted in FIG. 2E above), made from 83 feet of 14 AWG wire, and driven at 6.78 MHz. The rectangular loop with the range of aspect ratios shown in FIG. 3E represents various shapes the wire loop (204) shown in FIG. 2D may be adapted by the user to fit any wireless power transfer area. The plot shows the inductance of the rectangular loop as the perimeter (i.e., corresponding to the circumference of the wire loop (204)) is held fixed but the aspect ratio is varied. As can be seen from the plot, the inductance varies less than 20% as the aspect ratio is varied over a wide range between 0.05 and 0.95. Accordingly, the characteristic frequency of the wire loop (204) varies less than 10% while being adapted into a rectangular loop over a wide range of aspect ratios. This demonstrates the relative insensitivity of the resonant frequency of the loop with distributed capacitance to variations in the adapted form factor.

Referring back to the discussion of FIG. 1A, in one or more embodiments of the invention, the system (100) provides wireless power transfer across the wireless power transfer area (101) based on the ISM band. In the scenario where the variable form factor transmitter (102) is approximated by the parallel-wire transmission line (201) shown in FIG. 2A, 2B, or 2C, the values of the wire segment length s, the inductance per unit length l, and the capacitor C may be chosen in the factory, based on Eq. (9b), to maintain the resonant angular frequency ω_(o) of the parallel-wire transmission line (201) equal to the angular frequency of the RF power source, which may be within the type A frequency range (i.e., 6.765 MHz-6.795 MHz) defined in the ITU Radio Regulations Article 5, footnote 5.138.

In the scenario where the variable form factor transmitter (102) is approximated by the wire loop (204) shown in FIG. 2D, the values of the wire segment length, s, and the capacitor, C, may be chosen in the factory, based on Eq. (12c), to maintain the resonant angular frequency ω_(o) of the wire loop (204) equal to the angular frequency of the RF power source, which may be within the type A frequency range (i.e., 6.765 MHz-6.795 MHz) defined in the ITU Radio Regulations Article 5, footnote 5.138.

In one or more embodiments of the invention, the aforementioned manufacturing tolerance is controlled such that the resulting capacitance range, length range, and inductance range do not cause the resonant angular frequency ω_(o) to deviate from the type A frequency range (i.e., 6.765 MHz-6.795 MHz). In addition for both scenarios described above, approximation error exists due to physical difference between the user adapted form factor of the variable form factor transmitter (102) and the simplified form factor of the parallel-wire transmission line (201) or the wire loop (204). In one or more embodiments of the invention, to compensate for the aforementioned manufacturing tolerance and the approximation error, the input impedance and the characteristic frequency of the variable form factor transmitter (102) may be adjustable in the factory as well as by the user.

Further to the discussion of FIG. 1A above, FIG. 2H shows schematic diagrams of an equivalent circuit A (205 a) and an equivalent circuit B (205 b) of the variable form factor transmitter (102). For optimal power transfer from the power source (108), the input impedance of the variable form factor transmitter (102) is matched to the output impedance (represented by the resistor R_(L)) of the power source (108). The resistor R is an effective series resistance representing all sources of loss (e.g., ohmic loss, radiation loss, dielectric loss, etc.) of the variable form factor transmitter (102). The variable capacitor C₁ determines the apparent input impedance of the variable form factor transmitter (102) at its resonant frequency, while the variable capacitor C₂ sets the resonant frequency.

The equivalent circuit B (205 b) corresponds to a simplified form of the equivalent circuit A (205 a) where C₂, C₃, and L have been combined into a single reactance, χ. The input impedance of the variable form factor transmitter (102) is equal to R_(L) when C₁ has the value given by Eq. (15).

$\begin{matrix} {C_{1} = {\frac{1}{\omega_{o}R_{L}}\sqrt{\frac{R_{l}}{R} - 1}}} & {{Eq}.\mspace{14mu}(15)} \end{matrix}$

For the case where R_(L)<R, the transformer coupling scheme shown in FIG. 2E may be used. For the case where R_(L)≥R, the capacitive coupling scheme shown in FIG. 2F may be used.

FIG. 2J shows example constructions of the variable form factor transmitter (102), depicted based on the legend (221), in accordance with one or more embodiments. In one or more embodiments, one or more of the modules and elements shown in FIG. 2J may be omitted, repeated, and/or substituted. Accordingly, embodiments of the invention should not be considered limited to the specific arrangements of modules shown in FIG. 2J.

As shown in FIG. 2J, the distributed capacitor string A (210 a) and distributed capacitor string B (210 b) are two example constructions of the variable form factor transmitter (102) depicted in FIGS. 1.1 and 1.2 above. Accordingly, the wire loop (204) and the rectangular loop (206), respectively depicted in FIGS. 2D and 2E above, may be based on the distributed capacitor string A (210 a), the distributed capacitor string B (210 b), or a combination thereof. In particular, the distributed capacitor string A (210 a) and distributed capacitor string B (210 b) show two example constructions corresponding to a cross-sectional view of the wire loop (204) or the rectangular loop (206). Specifically, the cross-sectional view includes a cross-section of consecutive capacitors and wire-segments in the wire loop (204) or the rectangular loop (206).

The distributed capacitor string A (210 a) includes capacitors constructed from conductive strips (230 a), (230 b), (230 c), (230 d), (230 e), (230 f), etc. attached to two opposing surfaces (i.e., surface A (220 a), surface B (220 b)) of a dielectric material sheet. Specifically, a sheet has a three-dimensional (3D) form factor with the majority (e.g., greater than 90%) of surface area occupied by the two opposing surfaces. In other words, the thickness (i.e., distance between the two opposing surfaces) of the sheet is substantially less than each dimension of the two opposing surfaces. In this context, the 3D form factor of the sheet may be represented as a two-dimensional (2D) form factor with the thickness along a third dimension perpendicular to a surface (e.g., surface A (220 a), surface B (220 b)) of the 2D form factor. A conductive strip is a sheet of conductive material that is attached to, and has a substantially smaller (e.g., less than 10%) area than, the dielectric material sheet. The distributed capacitor string A (210 a) is depicted in a cross-sectional view showing cross-sections of the conductive strips and the dielectric material sheet. In particular, the cross-sectional view cuts across the surface A (220 a) and surface B (220 b) along the third dimension to show the thicknesses of the conductive strips and the dielectric material sheet.

In one or more embodiments, one or more of the conductive strips (230 a), (230 b), (230 c), (230 d), (230 e), (230 f), etc. are printed on the surface A (220 a) and/or surface B (220 b) using conductive ink, paste, paint, or other conductive coating material. In one or more embodiments, one or more of the conductive strips (230 a), (230 b), (230 c), (230 d), (230 e), (230 f), etc. are formed by selectively etching one or more conductive films laminated with the dielectric material sheet. For example, the conductive film(s) and the dielectric material sheet may be laminated together by heat, pressure, adhesive, welding, or other suitable method.

For example, the capacitor (211) includes overlapping portions of the conductive strip (230 a) and conductive strip (230 e), respectively attached to the surface A (220 a) and surface B (220 b), that are separated by a thickness d of the dielectric material sheet. The overlapping portions of the conductive strip (230 a) and conductive strip (230 e) form two electrodes in a parallel-plate configuration of the capacitor (211). Similarly, the capacitor (213) includes overlapping portions of the conductive strip (230 b) and conductive strip (230 e), respectively attached to the surface A (220 a) and surface B (220 b), that are separated by the thickness d of the dielectric material sheet. The overlapping portions of the conductive strip (230 b) and conductive strip (230 e) form two electrodes in the parallel-plate configuration of the capacitor (213). The overlapping portions of two adjacent conductive strips are referred to as the overlap region having a distance x. Further, each of the conductive strips (230 a), (230 b), (230 c), (230 d), (230 e), (230 f), etc. acts as an inductive segment that connects two adjacent capacitors in the distributed capacitor string A (210 a). For example, the conductive strip (230 e) acts as, or otherwise implements, the inductive segment (212) to connect the capacitor (211) and capacitor (213) in series. Accordingly, the capacitor (211) and the inductive segment (212) form one of the multiple capacitor-wire segments of the distributed capacitor string A (210 a). Similarly, the capacitor (213) and the inductive segment (214) form another one of the multiple capacitor-wire segments of the distributed capacitor string A (210 a). In the distributed capacitor string A (210 a), the surface of the dielectric material sheet, where the inductive segments are attached, alternates between the surface A (210 a) and surface B (210 b). For example, the inductive segment (212) and one electrode of the capacitor (213) are integrated as a single conductive strip (230 e) attached to the surface A (210 a), while the inductive segment (214) and the other electrode of the capacitor (213) are integrated as a single conductive strip (230 b) attached to the opposing surface B (210 b). In this context, each capacitor-wire segment shown in the distributed capacitor string A (210 a) is a first type of integrated capacitor-wire segment. As used herein, the integrated capacitor-wire segment is a capacitor and an inductive segment that are connected in series where the inductive segment and one electrode of the capacitor are integrated into a single conductive strip.

Further as shown in FIG. 2J, the distributed capacitor string B (210 b) includes capacitors constructed from conductive strips (230 g), (230 h), (230 j), (230 k), (230 m), (230 n), (230 p), etc. attached to two opposing surfaces (i.e., surface C (220 c), surface D (220 d)) of a dielectric material sheet. Similar to the distributed capacitor string A (210 a), the distributed capacitor string B (210 b) may be constructed by printing, lamination, etching, or combinations thereof. For example, the capacitor (215) includes overlapping portions of the conductive strip (230 g) and conductive strip (230 m), respectively attached to the surface C (220 c) and surface D (220 d), that are separated by the thickness d of the dielectric material sheet. The overlapping portions of the conductive strip (230 g) and conductive strip (230 m) form two electrodes in the parallel-plate configuration of the capacitor (215). The capacitor (216) includes overlapping portions of the conductive strip (230 h) and conductive strip (230 m), respectively attached to the surface C (220 c) and surface D (220 d), that are separated by the thickness d of the dielectric material sheet. The overlapping portions of the conductive strip (230 h) and conductive strip (230 m) form two electrodes in the parallel-plate configuration of the capacitor (216). The capacitor (215) and capacitor (216) are connected together in series at the conductive strip (230 m) to form a combined capacitor (222) that is itself connected between the conductive strip (230 g) and conductive strip (230 h). Similarly, the combined capacitor (223), including two capacitors connected in series, is connected between the conductive strip (230 h) and conductive strip (230 j). Further, each of the conductive strips (230 g), (230 h), (230 j), etc. acts as an inductive segment that connects two adjacent combined capacitors in the distributed capacitor string B (210 b). For example, the conductive strip (230 h) acts as, or otherwise implements, the inductive segment (218) to connect the combined capacitor (222) and combined capacitor (223) in series. Accordingly, the combined capacitor (222) and the inductive segment (218) form one of the multiple capacitor-wire segments of the distributed capacitor string B (210 b). Similarly, the combined capacitor (223) and the inductive segment (219) form another one of the multiple capacitor-wire segments of the distributed capacitor string B (210 b). In the distributed capacitor string B (210 b), the inductive segments (215), (218), (219), etc. are attached to a single surface (i.e., surface C (220 c)) of the dielectric material sheet. For example, the inductive segment (218) and one electrode of the combined capacitor (223) are integrated as a single conductive strip (230 h) attached to the surface C (210 c), while the inductive segment (219) and the other electrode of the combined capacitor (223) are integrated as a single conductive strip (230 j) attached to the same surface C (210 c). In this context, each capacitor-wire segment shown in the distributed capacitor string B (210B) is a second type of integrated capacitor-wire segment.

As noted above, the wire loop (204) and the rectangular loop (206), respectively depicted in FIGS. 2D and 2E above, may be based on the distributed capacitor string A (210 a), the distributed capacitor string B (210 b), or a combination thereof. In other words, the first type of integrated capacitor-wire segment(s) in the distributed capacitor string A (210 a) and/or the second type of integrated capacitor-wire segment(s) in the distributed capacitor string B (210 b) may be included in the wire loop (204) and/or the rectangular loop (206), respectively depicted in FIGS. 2D and 2E above. Although a specific number of integrated capacitor-wire segments are shown in the distributed capacitor string A (210 a) and the distributed capacitor string B (210 b) above, the wire loop (204) and/or the rectangular loop (206) may also include more number of integrated capacitor-wire segments of either type, or less number of integrated capacitor-wire segments of either type, than what is shown in the distributed capacitor string A (210 a) and the distributed capacitor string B (210 b). In one or more embodiments, either type or both types of integrated capacitor-wire segments may be combined with other forms of capacitor-wire segments (e.g., based on discrete capacitor(s) and inductor(s)) to form the wire loop (204) and/or the rectangular loop (206), respectively depicted in FIGS. 2D and 2E above.

FIG. 2K shows an example construction of the variable form factor transmitter (102), depicted based on the legend (221), in accordance with one or more embodiments. In one or more embodiments, one or more of the modules and elements shown in FIG. 2K may be omitted, repeated, and/or substituted. Accordingly, embodiments of the invention should not be considered limited to the specific arrangements of modules shown in FIG. 2K.

As shown in FIG. 2K, the power transmitter (250), depicted based on the legend (221), shows an example construction corresponding to a three-dimensional (3D) view of FIG. 2E above. The rectangular loop (206) of the power transmitter (250) is based on the distributed capacitor string A (210 a) described in reference to FIG. 2J above. Specifically, a portion (224) of the rectangular loop (206) corresponds to a 3D view of the distributed capacitor string A (210 a) depicted in FIG. 2J. In other words, the distributed capacitor string A (210 a) depicted in FIG. 2J corresponds to a cross section (designated by the double arrowed and dashed line) of the portion (224). The thicknesses of the conductive strips and the dielectric material sheet in the distributed capacitor string A (210 a) are omitted in the 3D view for clarity of showing the aforementioned 2D form factors.

RF characteristics of the rectangular loop (206) is described below, where the width, length, and number of overlap regions of the rectangular loop (206) are denoted as a, b, and n, respectively. The overlap area A between two adjacent conductive strips may be computed using Eq. (16), where the conductive strip width, conductive strip length, and overlapping region distance are denoted as w, l, and x, respectively. A=w×x  Eq. (16)

The capacitance of each overlap region may be computed using Eq. (17), where the dielectric constant and the thickness of the dielectric material sheet, are denoted as ε and d, respectively.

$\begin{matrix} {C = \frac{ɛ\; A}{d}} & {{Eq}.\mspace{14mu}(17)} \end{matrix}$

The total capacitance of the rectangular loop (206) may be computed using Eq. (18).

$\begin{matrix} {C_{tot} = \frac{C}{n}} & {{Eq}.\mspace{14mu}(18)} \end{matrix}$

The total inductance of the rectangular loop (206) may be computed using Eq. (19) and Eq. (20).

$\begin{matrix} \left. {L = {\frac{\mu}{\pi}\left\lbrack {{a\mspace{11mu}\ln\frac{2\; a}{\rho}} + {b\mspace{11mu}\ln\frac{2\; b}{\rho}} + {2\sqrt{a^{2} + b^{2}}} - {a\mspace{11mu}\sinh^{- 1}\frac{a}{b}} - \;{b\;\sinh^{- 1}\frac{b}{a}} - {2\left( {a + b} \right)}} \right)}} \right\rbrack & {{Eq}.\mspace{14mu}(19)} \\ {\mspace{79mu}{\rho = \frac{\omega}{4}}} & {{Eq}.\mspace{14mu}(20)} \end{matrix}$

The resonant frequency ω_(o) of the rectangular loop (206) may be computed using Eq. (21).

$\begin{matrix} {\omega_{o} = \frac{1}{\sqrt{{LC}_{tot}}}} & {{Eq}.\mspace{14mu}(21)} \end{matrix}$

Additional relationships between the resonant frequency and other parameters of the rectangular loop (206) include Eq. (22), Eq. (23), and E1. (24).

$\begin{matrix} {C_{tot} = \frac{1}{L\;\omega_{o}^{2}}} & {{Eq}.\mspace{14mu}(22)} \\ {\frac{ɛ\; A}{d} = \frac{n}{L\;\omega_{o}^{2}}} & {{Eq}.\mspace{14mu}(23)} \\ {\frac{A}{d} = \frac{n}{ɛ\; L\;\omega_{o}^{2}}} & {{Eq}.\mspace{14mu}(24)} \end{matrix}$

TABLE 2 lists four examples of the RF characteristics of the rectangular loop (206) based on the equations above.

TABLE 2 d (nm) A (mm²) n ε (F/m) L (H) ω_(o) ² (Hz²) 894.5 63.5 20 8.854 E−12 1.753 E−6 1.8148 E15 100.3 6.35 20 8.854 E−12 1.753 E−6 1.8148 E15 767.3 63.5 10 8.854 E−12 1.753 E−6 1.8148 E15 8.945 63.5 20 8.854 E−12 1.753 E−6 1.8148 E13

In one or more embodiments, the power transmitter (250) is configured based on a pre-determined wireless power transfer area. For example, the pre-determined wireless power transfer area may be a table top surface where one or more mobile receiver devices (e.g., mobile phone) are placed to receive wireless power transfer. The rectangular loop (206) may be movably or permanently disposed along a path based on the table top surface. For example, the path may be the edges of the table top surface, on top of or beneath the table top surface, on a fixture or ceiling above the table top surface, on or embedded in the floor below the table top surface, etc. The power source (108) is connected to the capacitor-wire segments via the terminal A (202 a) and terminal B (202 b), and may be plugged into a power outlet on a wall near the table top surface. The dielectric material sheet (225) encompasses at least a portion of the path to implement the capacitor(s) of the rectangular loop (206) and to provide mechanical support for the rectangular loop (206).

In another example, the pre-determined wireless power transfer area may be a space adjacent to a window where one or more receiver devices (e.g., mobile phone) are disposed about the space to receive wireless power transfer. The rectangular loop (206) may be movably or permanently disposed along a path based on the window. For example, the path may be the edges of the window frame, in front of or behind the window glass surface, embedded in the window glass or the window frame, etc. The power source (108) may be plugged into a power outlet on a wall where the window is mounted, or wired to the power outlet behind the surface of the wall.

In one or more embodiments, one or more of the dielectric material sheet, conductive strips, and/or integrated capacitor-wire segments may be rigid or pliable, transparent, translucent, or opaque depending on respective thicknesses and/or compositions. Although 2D form factors of the dielectric material sheet, conductive strips, and/or integrated capacitor-wire segments are shown as rectangular shapes in FIG. 2K, different 2D form factors (e.g., polygonal, circular, oval, elliptical, spiral, etc. shapes or combinations thereof) than what is shown may also be exhibited by the dielectric material sheet, conductive strips, and/or integrated capacitor-wire segments. Although the conductive strips in the rectangular loop (206) follows a path outlining a rectangular shape in FIG. 2K, the conductive strips in the rectangular loop (206) may also follow a different path outlining a different shape that turns the rectangular loop (206) into a loop with a different shape, such as a polygonal, circular, oval, elliptical, spiral, etc. loop or combinations thereof. Although a specific number of conductive strips and/or integrated capacitor-wire segments in the rectangular loop (206) are shown in FIG. 2K, the power transmitter (250) may also include more number of integrated capacitor-wire segments, or less number of integrated capacitor-wire segments than what is shown.

FIG. 2L shows an example construction of the variable form factor transmitter (102) in accordance with one or more embodiments. In one or more embodiments, one or more of the modules and elements shown in FIG. 2L may be omitted, repeated, and/or substituted. Accordingly, embodiments of the invention should not be considered limited to the specific arrangements of modules shown in FIG. 2L.

As shown in FIG. 2L, the power transmitter (260), depicted based on the legend (231), shows an example construction corresponding to a top view of the power transmitter (250) depicted in FIG. 2K above. In particular, the top view has a viewing direction along the aforementioned third dimension perpendicular to the surface (e.g., surface A (220 a), surface B (220 b)) of the power transmitter (250). The power transmitter (260) includes a rectangular loop (206 a) implemented using the dielectric material sheet (251) and connected to the power source (108). The rectangular loop (206 a) and dielectric material sheet (251) are variations of the rectangular loop (206) and dielectric material sheet (225) depicted in FIG. 2K above. For, example, the rectangular loop (206 a) and the rectangular loop (206) have different number of capacitor-wire segments. Further, the dielectric material sheet (251) includes an opening (233) where the dielectric material is cut out from the dielectric material sheet (251).

In one or more embodiments, the power source (108) is implemented using at least a flexible circuit having a thin insulating polymer film with conductive circuit patterns and electronic chips affixed thereto. For example, the flexible circuit may be attached to and/or mechanically supported by the dielectric material sheet (251). Example details of a portion of the power transmitter (260) containing the power source (108) is shown in FIG. 2M below based on the legend (241).

As shown in FIG. 2M, the power source (108) includes a flexible circuit (108 a) connected to several conductive strips attached to a surface of the dielectric material sheet (251). The conductive strips include a spiral A (209 a), a spiral B (209 b), a spiral C (209 c), and a spiral D (209 d). One end of the spiral A (209 a) is designated as the terminal A (204 a), one end of the spiral B (209 b) is designated as the terminal B (204 b), one end of the spiral C (209 c) is designated as the terminal C (204 c), and one end of the spiral D (209 d) is designated as the terminal D (204 d). The other ends of the spiral A (209 a) and spiral B (209 b) are connected together using the conductive bridge A (209 d) to implement a secondary winding of an isolation transformer contained in the power source (108). The other ends of the spiral C (209 c) and spiral D (209 d) are connected together using the conductive bridge B (209 e) to implement a primary winding of the isolation transformer. The conductive bridge A (209 d) and conductive bridge B (209 e) may be implemented using insulated conductive wires or other electrical connection means. The primary and secondary windings intertwine with each other to provide an inductance coupling effect of the isolation transformer. In addition, certain capacitors (designated as C1 and C2) contained in the power source (108) may be connected to the terminal A (204 a), terminal B (204 b), terminal C (204 c), and terminal D (204 d). The capacitors C1 and C2 may be discrete capacitors soldered to the terminals or capacitors implemented using additional conductive strips attached to the two opposing surfaces of the dielectric material sheet (251). For example, the isolation transformer and the capacitors C1 and C2 may be part of or related to an impedance matching circuit to substantially match a pre-determined output impedance of the power source (108) to the string of distributed capacitors in the rectangular loop (206 a) depicted in FIG. 2L.

FIG. 2N shows an application example of a wireless power transfer area, based on the power transmitter (250) depicted in FIG. 2K above, in accordance with one or more embodiments. In one or more embodiments, one or more of the modules and elements shown in FIG. 2N may be omitted, repeated, and/or substituted. Accordingly, embodiments of the invention should not be considered limited to the specific arrangements of modules shown in FIG. 2N.

As shown in FIG. 2N, the wireless power transfer area includes the table top (600) where the rectangular loop (206) of the power transmitter (250) follows the edges of the table top (600). The dielectric material sheet (225) is laid on top of the table top (600) where the thickness is omitted for clarity. The power cord and plug of the power source (108) are also omitted. The receiver device A (500 a) and receiver devices B (500 b) receive wireless power transfer from the power transmitter (250) to light up a string of decorative light emitting diodes (LEDs) attached to the bottoms of glasses. Examples of the receiver device A (500 a) and receiver devices B (500 b) are described in reference to FIGS. 5A, 5B, 5D, and 5E below. In addition, the receiver device C (500 c) is a commercially available product that receives wireless power transfer from the power transmitter (250) to charge the battery of a mobile device (500), such as a mobile phone, tablet computer, notebook computer, etc. TABLE 3 shows input power and input current for four example loading scenarios of the power transmitter (250).

TABLE 3 Input power (W) Input current (A) Number of Load 23.23 1.96 0 23.80 2.01 1 LED 24.82 2.10 2 LED 25.63 2.17 3 LED 28.38 2.17 3 LED + battery charger

Similar to the distributed capacitor string A (210 a) and distributed capacitor string B (210 b) depicted in FIG. 2J above, FIG. 2P shows additional constructions of the variable form factor transmitter in accordance with one or more embodiments. In particular, overlapping portions of the two conducting films correspond to capacitors while non-overlapping portions of either conducting film correspond to inductors. In one or more embodiments, one or more of the modules and elements shown in FIG. 2P may be omitted, repeated, and/or substituted. Accordingly, embodiments of the invention should not be considered limited to the specific arrangements of modules shown in FIG. 2P.

As shown in FIG. 2P, the distributed capacitor string C (210 c), distributed capacitor string D (210 d), and distributed capacitor string E (210 e) are three additional example constructions of the variable form factor transmitter (102) depicted in FIGS. 1.1 and 1.2 above. According to the legend (300), the dielectric is a layer which separates two layers of conducting film. The mechanical support is not provided by the dielectric, but rather by a separate and distinct insulating mechanical substrate that is shown below both the conducting film layers and the dielectric layer. The dielectric may include an oxide layer grown on the surface of one of the layers of conducting film, which may be a metallic conductor. The dielectric layer may cover the entire upper surface of the lower conducting layer, as shown in the distributed capacitor string C (210 c), or may only cover the region of overlap, as shown in the distributed capacitor string D (210 d).

Alternatively, as shown in the distributed capacitor string E (210 e), the dielectric may consist of a thin film of insulating material to which both the upper and lower conducting films adhere. However, the dielectric may be too thin to provide adequate mechanical support, in which case all three layers may be superposed above an additional insulating layer, which provides mechanical support of the dielectric and upper/lower conducting films.

FIG. 4A shows a schematic diagram of an example RF power source in accordance with one or more embodiments of the invention. In particular, the example RF power source (108) shown in FIG. 4A may operate based on the ISM band as the power source (108) depicted in FIGS. 1A, 1C, 2B, 2C, 2D, 2K, 2L, and 2M above. Specifically, the example RF power source (108) shown in FIG. 4A includes the terminal A (204 a) and terminal B (204 b) that correspond to the two terminals of the power source (108) depicted in FIGS. 1A, 1C, 2B, 2C, 2D, 2K, 2L, and 2M above. The schematic diagram includes capacitors, inductors, and resistors of various RLC circuit components and commercial part numbers of various integrated circuit components. In particular, the inductors designated as L1 and L2 correspond to the primary conductive winding and secondary conductive winding shown in FIG. 2M above. The capacitors designated as C1 and C2 correspond to like-named capacitors shown in FIG. 2M above. In one or more embodiments, one or more of the modules and elements shown in FIG. 4A may be omitted, repeated, and/or substituted. Accordingly, embodiments of the invention should not be considered limited to the specific arrangements of modules shown in FIG. 4A.

FIG. 4B shows a schematic diagram of an example RF power source connected to an equivalent circuit in accordance with one or more embodiments of the invention. In particular, the example RF power source (108) shown in FIG. 4B may operate based on the ISM band as the power source (108) depicted in FIGS. 1A, 1C, 2B, 2C, 2D, 2K, and 2L above. Specifically, the example RF power source (108) shown in FIG. 4B includes the terminal A (204 a) and terminal B (204 b) that correspond to the two terminals of the power source (108) depicted in FIGS. 1A, 1C, 2B, 2C, 2D, 2K, and 2L above. The schematic diagram includes capacitors, inductors, and resistors of various RLC circuit components and commercial part numbers of various integrated circuit components. In particular, the equivalent circuit (206 b) represents the rectangular loop (206) or rectangular loop (206 a) shown in FIGS. 2K and 2L above. In one or more embodiments, one or more of the modules and elements shown in FIG. 4B may be omitted, repeated, and/or substituted. Accordingly, embodiments of the invention should not be considered limited to the specific arrangements of modules shown in FIG. 4B.

FIG. 5A shows a schematic diagram of an example receiver device A (500 a) in accordance with one or more embodiments of the invention. In one or more embodiments, one or more of the modules and elements shown in FIG. 5A may be omitted, repeated, and/or substituted. Accordingly, embodiments of the invention should not be considered limited to the specific arrangements of modules shown in FIG. 5A.

As shown in FIG. 5A, the receiver device A (500 a) includes multiple light emitting diodes (LEDs) (e.g., LED (502)) that are connected in parallel to form an LED string. The two ends of the LED string are connected to a rectifier circuit A (501 a) to form a loop. For example, the loop may be a circular loop used as a mobile LED lighting device used within the wireless power transfer area (101) depicted in FIG. 1A above. In one or more embodiments of the invention, the rectifier circuit A (501 a) includes capacitors C₁, C₂, and C₃ and rectifying diodes D₁ and D₂. When the receiver device A (500 a) is in the presence of the oscillating magnetic fields, the changing magnetic flux through the loop of the LED string induces a voltage difference between the two ends of the LED string. The induced voltage difference oscillates with time. The capacitance C₃ is adjusted to bring the LED string into resonance with the oscillating magnetic fields to enhance the induced oscillating voltage. The rectifying diodes D₁ and D₂ rectify the induced oscillating voltage to produce a DC voltage difference between the outer wire (503 a) and inner wire (503 b) of the LED string thereby deliver power to the parallel-connected LEDs (e.g., LED (502)). The capacitors C₁ and C₂ act as RF bypass capacitors to maintain the outer wire (503 a) and inner wire (503 b) of the LED string appear shorted to the RF current. The configuration of the receiver device A (500 a) limits the loop voltage by the combined forward voltage drop across the LEDs in series with the rectifying diode D₁ or D₂, which improves safety to the user.

Similar to FIG. 5A, FIG. 5B shows an example receiver device B (500 b), which is a larger version of the receiver device A (500 a) that has multiple rectifier circuits (i.e., rectifier circuit B (501 b), rectifier circuit C (501 c), rectifier circuit D (501 d), rectifier circuit E (501 e)). The operation of the receiver device B (500 b) is substantially the same as the receiver device A (500 a). The number of segments in the receiver device B (500 b) may be chosen to provide an optimal impedance match to the load, i.e., the parallel-connected LEDs.

In addition to FIGS. 5A and 5B, FIG. 5C shows a schematic diagram of other example receiver devices.

FIG. 5C shows a schematic diagram of an example receiver device circuit (500 c) in accordance with one or more embodiments of the invention. In one or more embodiments, the receiver device circuit (500 c) is employed in various types of receiver devices having different shapes, sizes, form factors, etc. for various different types of mobile or stationery applications within the wireless power transfer area (101) depicted in FIG. 1A above. In one or more embodiments, at least the inductor, L, of the receiver device circuit (500 c) is placed within the wireless power transfer area (101) for receiving the wireless power transfer. The remaining components shown in FIG. 5C are configured to convert the received wireless power to suitable format to be consumed by a load, represented by the resistance, R_(L).

As shown in FIG. 5C, the inductor, L, along with capacitors, C₁, C₂, and C₃, are tuned to resonate at the characteristic frequency of the variable form factor transmitter (102) and the RF power source (108) described in reference to FIGS. 1A through 2G above. The value of capacitor C₁ is chosen to provide an impedance match between the resonant receiver and the input of the DC-to-DC converter (504). The DC-to-DC converter (504) transforms the rectified voltage into a constant voltage to drive the load, R_(L). The DC-to-DC converter (504) allows the receiver device circuit (500 c) to present a constant voltage to the load R_(L) even in situations where the receiver device circuit (500 c) is moved through regions of varying magnetic field strength within the wireless power transfer area (101). Note that the load R_(L) need not be a linear device, i.e., a device with a linear voltage versus current relation. Examples of load R_(L) include, but are not limited to, LED's, microcontrollers, motors, sensors, actuators, etc.

FIG. 5D shows a schematic diagram of an additional example receiver device circuit (500 d) in accordance with one or more embodiments of the invention. The inductor, L, along with capacitors, C₁ and C₂, are tuned to resonate at the characteristic frequency of the variable form factor transmitter (102) and the RF power source (108) described in reference to FIGS. 1A through 2G above. The value of capacitor C₁ is chosen to provide an impedance match between the resonant receiver and the LED load. The bridge rectifier converts the RF voltage present on capacitor C₁ into a DC voltage, which drives the LED. For example, the LED may correspond to the string of decorative light emitting diodes (LEDs) attached to the bottoms of glasses, depicted in FIG. 2N above.

FIG. 5E shows a layout diagram (500 e) of the example receiver device circuit (500 d) depicted in FIG. 5D above. The inductor, L, is composed of a conducting trace on the surface of a printed circuit board (PCB) in the form of a flat spiral with multiple turns. Capacitors, C₁ and C₂, are placed in series with this spiral at the location (501). A second layer of traces is used on the PCB to allow connections to jump over multiple turns of the inductor, L. Note also that C₁ and C₂ can be placed in series with the turns of the inductor, L, at any point. In FIG. 5E, for example, the capacitor, C₂, is placed across a break in the center of the inductor, L. This placement helps to maintain symmetry in the distribution of voltage on the turns of the inductor, L.

In one or more embodiments of the invention, the receiver device A (500 a), receiver device B (500 b), receiver device circuit (500 c), or receiver device circuit (500 d) may receive power wirelessly from any electromagnetic transmitter, such as a dipole transmitter (e.g., magnetic dipole transmitter), a loop antenna with distributed capacitance, a parallel-wire transmission line with distributed capacitance, a shielded transmission line with distributed capacitance, etc. In one or more embodiments of the invention, the receiver device A (500 a), receiver device B (500 b), receiver device circuit (500 c), and/or or receiver device circuit (500 d) are placed within the wireless power transfer area (101) as the receiver device (A), receiver device (B), receiver device (C), receiver device (D), receiver device (E), or receiver device (F) to receive power wirelessly from the variable form factor transmitter (102).

FIGS. 6A-6F show schematic and layout diagrams, according to legend (600), of variations of the variable form factor transmitter (102) depicted in FIGS. 1A-1C above. In one or more embodiments of the invention, the variable form factor transmitter (102) shown in FIGS. 6A-6F may include, or otherwise be based on, example constructions described in reference to FIGS. 2A-2P above. For clarity, the distributed capacitors may not be explicitly shown in FIGS. 6A-6F. In one or more embodiments, the wireless power transfer area (101) shown in FIGS. 6A-6F may be a large wireless power transfer area. A large wireless power transfer area, or simply large area, has a dimension (e.g., length, width, diameter, etc.) that exceeds a wavelength corresponding to the aforementioned characteristic frequency of the variable form factor transmitter (102).

In one or more embodiments of the invention, the variable form factor transmitter (102) shown in FIGS. 6A-6F may be arranged in a 2-dimensional spatially-periodic structure to suppress far-field radiation due to oscillating current density. In particular, suppressing far-field radiation in the variable form factor transmitter (102) shown in FIGS. 6A-6F is described in reference to FIGS. 7A-7F below.

In one or more embodiments, one or more of the modules and elements shown in FIGS. 6A-6F may be omitted, repeated, and/or substituted. Accordingly, embodiments of the invention should not be considered limited to the specific arrangements of modules shown in FIGS. 6A-6F.

As shown in FIG. 6A, the variable form factor transmitter (102) includes a number of cross-coupled segments (e.g., segment A (601), segment B (601 b), segment C (601 c), etc.) disposed about the wireless power transfer area (101). As used herein, a segment is a contiguous portion of the variable form factor transmitter (102). For example, the segment may be a section of the distributed capacitor string (207) depicted in FIG. 2F above. In one or more embodiments, the segment is arranged into a loop form factor substantially enclosing a loop area and has at least one pair of terminals for electrical connection to an adjacent segment (referred to as a neighbor). Adjacent segments are two segments without any intervening segment disposed between them. In one or more embodiments, adjacent segments are electrically connected to each other to pass electrical current from one segment to the other segment. For example, the segment A (601) has one pair of terminals (602 a) connected to another pair of terminals (602 b) of the segment B (601 b). In another example, the segment B (601 b) has two pairs of terminals for connecting to two adjacent segments (i.e., segment A (601 a) and segment C (601 c)). In one or more embodiments, the electrical current flowing in each segment induces magnetic field substantially orthogonal to the loop area of the segment. The direction of the induced magnetic field is dependent on the rotational direction of the electrical current flowing through the segment. The rotational direction is one of clockwise direction and counter clockwise direction. In one or more embodiments, the current is alternating current with directions of all currents reversed during half of the cycle. The direction of the induced magnetic field is related to the rotational direction of the flowing current according to a right-hand rule of the electromagnetic theorem.

In one or more embodiments, the segments of the variable form factor transmitter (102) are configured to transmit, from a radio frequency (RF) power source and based at least in part on the aforementioned characteristic frequency, RF power across the wireless power transfer area (101) via a near electromagnetic field of the variable form factor transmitter (102).

In one or more embodiments, each segment includes one or more sides where adjacent sides of the adjacent segments are configured to conduct electrical current in opposing rotational directions. For example, the electrical currents flow in counter clockwise direction in the segment A (601 a) and segment C (601 c) such that the induced magnetic fields are in the magnetic field direction A, which corresponds to the direction flowing out of the loop area of the segment A (601 a) and segment C (601 c) toward the viewer of FIG. 6A. In contrast, the electrical current flows in clockwise direction in the segment B (601 b) such that the induced magnetic fields are in the magnetic field direction B, which corresponds to the direction flowing into the loop area of the segment B (601 b) away from the viewer of FIG. 6A. As a result of the opposing directions of magnetic fields induced by adjacent segments, a radiation loss of the wireless power transfer due to a far electromagnetic field of the variable form factor transmitter (102) is reduced. The repetitive pattern of opposing magnetic field directions (i.e., out-of-phase directions) in adjacent segments is referred to as a checkerboard phase pattern. FIG. 6A shows a one dimensional checkerboard pattern. Examples of two dimensional checkerboard patterns are shown in FIGS. 6E-6G below.

In one or more embodiments, the RF power source includes multiple phase-locked amplifiers disposed in at least a portion of the segments of the variable form factor transmitter (102). For example, each segment shown in FIG. 6A includes a phase-locked RF amplifier. In one or more embodiments, phase-locking among the segments is accomplished using a phase-locked loop and a master-slave topology, in which all phase-locked RF amplifier share a master clock signal. While FIG. 6A depicted connected segments forming a single loop, in an alternative configuration having disconnected but coupled loops, each loop may have a phase-locked RF amplifier that measures its phase difference from its neighbors, and adjusts its phase to be opposite to the mean phase of its nearest neighbors. In this case, the multiple phase-locked amplifiers of the variable form factor transmitter (102) automatically arrange themselves into a checkerboard pattern of alternating phases, without the need for any centralized control. As used herein, alternating phases refers to opposing rotational directions of electrical currents flowing in adjacent segments.

In one or more embodiments, the segments of the variable form factor transmitter (102) have same shapes and/or dimensions. In one or more embodiments, one or more segments of the variable form factor transmitter (102) may have different shapes and/or dimensions as compared to remaining segments in the variable form factor transmitter (102). In one or more embodiments, the segments of the variable form factor transmitter (102) are disposed in one dimensional or multi-dimensional repetitive structure.

FIG. 6B shows a schematic layout diagram of a variation of the example variable form factor transmitter (102) depicted in FIG. 6A above where the RF power source includes a single RF amplifier. As shown in FIG. 6B, each segment is associated with multiple magnetic direction symbols to illustrate substantially uniform magnetic field across the loop area.

FIG. 6C shows a schematic layout diagram of a variation of the example variable form factor transmitter (102) depicted in FIG. 6A above where the segments are partitioned into two separate portions denoted as variable form factor transmitter A (102 a) and variable form factor transmitter B (102 b) that are individually powered by separate RF amplifiers denoted as “A” and “B”. In one or more embodiments, the variable form factor transmitter A (102 a) and variable form factor transmitter B (102 b) are physically shifted by half a segment width, and driven with a 90 degree relative phase shift. In particular, this arrangement eliminates the nulls in the vertical component of the magnetic field which exist in the arrangement shown in FIG. 6B. The 90 degree phase shift does not have any effect on the radiation loss.

FIG. 6D shows a schematic layout diagram of a variation of the example variable form factor transmitter (102) depicted in FIG. 6C above where the variable form factor transmitter A (102 a) is not explicitly driven by an RF amplifier. Instead, the induced magnetic fields from the variable form factor transmitter B (102 b) in turn induces electric current to flow in the segments of the variable form factor transmitter A (102 a). Although the variable form factor transmitter A (102 a) and variable form factor transmitter B (102 b) are independent of each other in terms of physical and electrical connections, they become magnetically coupled resonators.

In the case where the variable form factor transmitter (102) is constructed from multiple magnetically coupled segments, the resonant properties of the combined structure may be used to ensure the correct phase relation exists between the magnetically coupled segments without the need for active phase control. The effect employed in this case is the frequency splitting which develops between any two or more coupled resonators. When the resonators are arranged in a periodic structure, the resonant eigen-frequencies approach the form of a continuous band as the size of the structure is extended without bound. Bloch's theorem applies to such a structure, and the eigen-modes of excitation are described by Bloch wavefunctions. The eigen-mode with the highest spatial frequency has the lowest temporal frequency. This highest spatial-frequency eigen-mode corresponds to the desired checkerboard phase pattern. It is therefore possible to ensure the correct phase relation between individual variable form factor transmitters by driving the structure at its lowest temporal eigen-frequency within the Bloch band.

In one or more embodiments, the wireless power transfer operates at a fixed, pre-determined frequency. The system of magnetically coupled segments therefore are designed such that the lowest temporal-frequency eigen-mode in the Bloch band has a resonant frequency equal to the desired, pre-determined frequency of the wireless power transfer.

The system of magnetically coupled segments may be driven by an amplifier connected to a single segment, in which case the RF power spreads through the structure to establish the checkerboard phase pattern described above, so long as the resonant conditions described above are met. However, if the magnetically coupled segments have any radiation loss, the amplitude of the magnetic field decays geometrically with every segment-to-segment hop. Depending on the size of the system, it may therefore be necessary to drive the system from multiple points. If this is the case, then each separate amplifier must be properly phase-locked, so as to maintain the desired checkerboard phase pattern.

Note that when the total length of conductive wire in a single segment approaches a half of the free-space wavelength of the frequency of operation, there will be substantial charge build-up on the wire structure due to its self-capacitance. The solution is to add distributed capacitors at regular intervals in series with the segment. In other words, each segment is constructed as a distributed capacitor string described above.

FIG. 6E shows a schematic layout diagram of a variation of the example variable form factor transmitter (102) depicted in FIG. 6A above where the segments have substantially rectangular shapes that are arranged to cover a large two dimensional geographic area. Note that the transmitter depicted in FIG. 6E consists of multiple, disconnected loops of wire which are inductively coupled. Like the transmitter shown in FIG. 6D, this inductive coupling ensures that power is spread evenly throughout the structure.

FIG. 6F shows a schematic layout diagram of a variation of the example variable form factor transmitter (102) depicted in FIG. 6E above where the segments are connected according to an endless knot pattern.

FIG. 6G shows a schematic layout diagram of a variation of the example variable form factor transmitter (102) depicted in FIG. 6D above where each segment has three neighbors (i.e., adjacent segments), in contrast to the four-neighbor configurations depicted in FIGS. 6E and 6F above.

FIGS. 7A-7F illustrates suppression of far-field radiation by an oscillating current density arranged in a 2-dimensional spatially-periodic structure. FIG. 7A shows a two-dimensional, spatially-periodic current distribution, lying entirely in the x-y plane. Let the vectors a₁ and a₂ denote the two primitive translation vectors of the structure.

The position-space current density function, J(x), has the property: J(x+n ₁ a ₁ +n ₂ a ₂)=J(x)  (1) where n₁ and n₂ are any integers. Equation 1 expresses the discrete translational symmetry of the spatially-periodic structure.

The reciprocal lattice vectors, b₁ and b₂, may be written as functions of the primitive translation vectors: b ₁≡2πa ₂·(a ₁ ∧a ₂)⁻¹  (2) b ₂≡−2πa ₁·(a ₁ ∧a ₂)⁻¹  (3) where {circumflex over ( )} denotes the wedge product of two vectors. The reciprocal lattice vectors have the property: b _(n) ·a _(m)=2πδ_(nm)  (4) where δ_(nm) is the Kronecker delta function.

Additionally, let it be stipulated that the current density has the additional symmetry:

$\begin{matrix} {{J\left( {{x \pm \frac{a_{1}}{2}} \pm \frac{a_{2}}{2}} \right)} = {- {J(x)}}} & (5) \end{matrix}$

The condition stipulated in Equation 5 ensures that the current density exhibits a checkerboard-pattern translational symmetry, in addition to the periodic translational symmetry described by Equation 1. Inspection of FIG. 7A will show that the current distribution depicted therein obeys the checkerboard-pattern translational symmetry described by Equation 5.

Let {tilde over (J)}(k) denote the spatial Fourier Transform of the current density, where k is the spatial wavevector.

FIG. 7B shows the reciprocal lattice in Fourier space. The function, {tilde over (J)}(k), will be zero everywhere, except on lines of singularity extended infinitely, parallel to the k_(z) axis, and centered on each of the lattice points of the reciprocal lattice in the k_(x)-k_(y) plane, as depicted in FIG. 7B.

The condition stipulated in Equation 5 implies the following condition in Fourier space:

$\begin{matrix} {{{\overset{\sim}{J}(k)}e^{{- {ik}} \cdot {({{\pm \frac{a_{1}}{2}} \pm \frac{a_{2}}{2}})}}} = {- {\overset{\sim}{J}(k)}}} & (6) \\ {\left. \Longrightarrow{\overset{\sim}{J}(k)} \right. = {\frac{1}{2}\left( {1 - e^{{- {ik}} \cdot {({{\pm \frac{a_{1}}{2}} \pm \frac{a_{2}}{2}})}}} \right){\overset{\sim}{J}(k)}}} & (7) \end{matrix}$

Equation 7 implies that {tilde over (J)}(k) must be zero at any point where the dot-product, k·(±a₁/2±a₂/2), is an integer multiple of 2π. At the lattice points of the reciprocal lattice, the wavevector, k, may be expressed as: k=n ₁ b ₁ +n ₂ b ₂  (8) where n₁ and n₂ are integers. Therefore, {tilde over (J)}(k) will be zero at every lattice point of the reciprocal lattice where the following condition holds:

$\begin{matrix} {{{\pm \frac{n_{1}}{2}} \pm \frac{n_{2}}{2}} \in {\mathbb{Z}}} & (9) \end{matrix}$ Condition 9 will be satisfied at all lattice points where the integers n₁ and n₂ are either both odd or both even. Therefore, the only lattice points in the reciprocal lattice on which {tilde over (J)}(k) can take a non-zero value are those where one of the two integers is odd, and the other is even.

FIG. 7C shows the points in the k_(x)-k_(y) plane at which {tilde over (J)}(k) may take a non-zero value. The function {tilde over (J)}(k) consists of lines of singularity, centered at these points in the k_(x)-k_(y) plane and extended infinitely along the k_(z) axis.

Assume that the current distribution, J(x), oscillates sinusoidally in time with angular frequency, ω. Let c represent the speed of light in the surrounding medium, and let κ=ω/c represent the free-space wavenumber of a propagating electromagnetic wave with angular frequency, ω. The power radiated by the current density, J(x), is given by the following integral performed in Fourier space:

$\begin{matrix} {P_{rad} = {\frac{{\zeta\kappa}^{2}}{32\;\pi^{2}}{\int{d\;\Omega\overset{\sim}{J}*{(k) \cdot {\overset{\sim}{J}(k)}}}}}} & (10) \end{matrix}$ where k=κ{circumflex over (r)}, where {circumflex over (r)} is the direction unit vector of the outgoing radiation, and the angular integral, ∫dΩ, is taken over all directions of {circumflex over (r)}. Note that Equation 10 is only valid for distributions of current which are divergence-free, i.e. ∇·J=0.

The integral in Equation 10 is performed over the surface of a sphere in Fourier space, of radius κ, centered on the origin. FIGS. 7D and 7E show the circular cross-section of this surface in the k_(x)-k_(y) plane for case (a) and case (b), respectively. In particular, FIG. 7D shows an example case (a) in which far-field radiation is suppressed. In case (a), the magnitude of the wavenumber, κ, is less than both |b₁| and |b₂|. In contrast, FIG. 7E shows an example case (b) in which far-field radiation is not suppressed. In case (b), the magnitude of κ is larger than one of |b₁| and |b₂|.

In case (a), the sphere of radius κ does not intersect any of the lines of singularity on which {tilde over (J)}(k) takes on a non-zero value. Therefore, the value of the function {tilde over (J)}(k) is zero everywhere on the surface of this sphere, and the integral of Equation 10 is exactly zero for case (a).

In case (b), some of the lines of singularity, where {tilde over (J)}(k) takes on a non-zero value, intersect the surface of the sphere of radius κ. Therefore, the integral of Equation 10 is non-zero for case (b).

Accordingly, the radiated power will be suppressed as long as the following condition is met: κ<min(|b ₁ |,|b ₂|)  (11)

Note that the checkerboard-pattern translational symmetry, expressed by Equation 5, ensures the suppression of far-field radiation which occurs when Equation 11 is satisfied. This is because the checkerboard-pattern translational symmetry forces {tilde over (J)}(k) to have zero amplitude at all of the lattice points defined by Equation 9, including the origin. If the amplitude of {tilde over (J)}(k) were not zero at the origin of the reciprocal lattice, then {tilde over (J)}(k) would have a line of singularity extending along the k_(z) axis, which would intersect a sphere centered on the origin, regardless of the radius of the sphere. Therefore, the integral in Equation 10 would be non-zero for all values of κ.

FIG. 7F shows an example of a rectangular checkerboard grid where the current density is arranged. The two primitive translation vectors are: a ₁ =we ₁ +he ₂  (12) a ₂ =−we ₁ +he ₂  (13) where w and h are the width and height of the rectangular cells, and where e₁ and e₂ are unit vector pointing in the x and y directions, respectively. The reciprocal lattice vectors are:

$\begin{matrix} {b_{1} = {{\frac{\pi}{\omega}e_{1}} + {\frac{\pi}{h}e_{2}}}} & (14) \\ {b_{2} = {{{- \frac{\pi}{\omega}}e_{1}} + {\frac{\pi}{h}e_{2}}}} & (15) \end{matrix}$

Both reciprocal lattice vectors have the same magnitude, given by:

$\begin{matrix} {{b_{1}} = {{b_{2}} = {\pi\sqrt{\frac{1}{\omega^{2}} + \frac{1}{h^{2}}}}}} & (16) \end{matrix}$

The condition for the suppression of far-field radiation can be expressed as:

$\begin{matrix} {{\kappa < {\min\left( {{b_{1}},{b_{2}}} \right)}} = {\pi\sqrt{\frac{1}{\omega^{2}} + \frac{1}{h^{2}}}}} & (17) \end{matrix}$

This condition may also be expressed in terms of the free-space wavelength, λ=2π/κ:

$\begin{matrix} {\lambda > \frac{2}{\sqrt{\frac{1}{\omega^{2}} + \frac{1}{h^{2}}}}} & (18) \end{matrix}$

In the case where h>>w, the rectangular grid approaches a zebra-stripe pattern, and the condition for radiation suppression becomes: λ>2w  (19)

FIG. 8 shows a method flowchart in accordance with one or more embodiments of the invention. In one or more embodiments, the method may be based on one or more variable form factor transmitters depicted in FIGS. 6A-6G above. One or more steps shown in FIG. 8 may be omitted, repeated, and/or performed in a different order among different embodiments of the invention. Accordingly, embodiments of the invention should not be considered limited to the specific number and arrangement of steps shown in FIG. 8.

Initially in Step 801, a variable form factor transmitter is adapted into at least a number of cross-coupled segments disposed about a pre-determined wireless power transfer area. The pre-determined wireless power transfer area includes a dimension exceeding a wavelength corresponding to a characteristic frequency of the variable form factor transmitter. In one or more embodiments, one or more cross-coupled segments is constructed using a string of distributed capacitors.

In Step 802, a number of phase-locked amplifiers are disposed in at least a portion of the cross-coupled segments as a radio frequency (RF) power source for wireless power transfer.

In Step 803, from the RF power source and based at least in part on a characteristic frequency, RF power is transmitted across the pre-determined wireless power transfer area via a near electromagnetic field of the variable form factor transmitter.

In Step 804, based on opposing directions of magnetic fields induced by adjacent cross-coupled segments, a radiation loss of the wireless power transfer due to a far electromagnetic field of the variable form factor transmitter is reduced.

In Step 805, receiver devices are disposed within the pre-determined wireless power transfer area to receive the RF power transfer. In particular, a portion of the RF power transmitted via the variable form factor transmitter from the RF power source is received by the receiver devices, where the characteristic frequency is substantially independent of a number or placement of the receiver devices.

While the invention has been described with respect to a limited number of embodiments, those skilled in the art, having benefit of this disclosure, will appreciate that other embodiments can be devised which do not depart from the scope of the invention as disclosed herein. Accordingly, the scope of the invention should be limited only by the attached claims. 

What is claimed is:
 1. A method for wireless power transfer, comprising: adapting a variable form factor transmitter, having a form factor changeable by a user, into at least a plurality of cross-coupled segments, each forming a respective transmitter loop, disposed about a pre-determined wireless power transfer area, the respective transmitter loops formed from a string of distributed capacitors having a plurality of capacitors connected in series with one another via a plurality of wire segments, wherein each of the plurality of capacitors comprises a pre-determined capacitance, and each of the plurality of wire segments comprises a pre-determined segment length and a pre-determined inductance per unit length, adjacent ones of the plurality of cross-coupled segments inducing respective magnetic fields having opposite direction to each other when driven together by a radio frequency (RF) power source; transmitting, from the RF power source and based at least in part on a characteristic frequency dependent on at least the pre-determined capacitance and the pre-determined inductance per unit length, RF power across the pre-determined wireless power transfer area via a near electromagnetic field of the variable form factor transmitter; and wherein, based on opposing directions of magnetic fields induced by adjacent cross-coupled segments of the plurality of cross-coupled segments, a radiation loss of the wireless power transfer is reduced due to a far electromagnetic field of the variable form factor transmitter.
 2. The method of claim 1, wherein each of the plurality of cross-coupled segments comprises a plurality of sides; and wherein adjacent sides of the adjacent cross-coupled segments are configured to conduct electrical current in opposing rotational directions.
 3. The method of claim 1, further comprising: disposing a plurality of phase-locked amplifiers in at least a portion of the plurality of cross-coupled segments as the RF power source.
 4. The method of claim 1, wherein the plurality of cross-coupled segments are disposed about the predetermined wireless power transfer area according to a pattern of endless knot wherein adjacent sides of the adjacent cross-coupled segments are electrically connected together in series.
 5. The method of claim 1, wherein the pre-determined wireless power transfer area comprises a dimension exceeding a wavelength corresponding to the characteristic frequency of the variable form factor transmitter.
 6. The method of claim 1, further comprising: disposing a plurality of receiver devices within the pre-determined wireless power transfer area; and wherein a portion of the RF power transmitted via the variable form factor transmitter from the RF power source is received by the plurality of receiver devices.
 7. The method of claim 1, wherein: the respective transmitter loop for each of the plurality of cross-coupled segments comprises a pair of terminals electrically connected to a pair of terminals of an adjacent one of the plurality of cross-coupled segments; and electrical current flowing through the respective transmitter loop for each of the plurality of cross-coupled segments induces a magnetic field substantially orthogonal to a loop area of the respective transmitter loop.
 8. A transmitter for wireless power transfer, comprising: a variable form factor transmitter having a form factor changeable by a user, adapted to form a plurality of cross-coupled segments, each forming a respective transmitter loop, disposed about a pre-determined wireless power transfer area, the respective transmitter loops formed from a string of distributed capacitors having a plurality of capacitors connected in series with one another via a plurality of wire segments, wherein each of the plurality of capacitors comprises a pre-determined capacitance, and each of the plurality of wire segments comprises a pre-determined segment length and a pre-determined inductance per unit length, adjacent ones of the plurality of cross-coupled segments inducing respective magnetic fields having opposite direction to each other when driven together by a radio frequency (RF) power source; wherein the plurality of cross-coupled segments are configured to: transmit, from the RF power source and based at least in part on a characteristic frequency dependent on at least the pre-determined capacitance and the pre-determined inductance per unit length, RF power across the pre-determined wireless power transfer area via a near electromagnetic field of the transmitter; and reduce, based on opposing directions of magnetic fields induced by adjacent cross-coupled segments of the plurality of cross-coupled segments, a radiation loss of the wireless power transfer due to a far electromagnetic field of the transmitter.
 9. The transmitter of claim 8, wherein each of the plurality of cross-coupled segments comprises a plurality of sides; and wherein adjacent sides of the adjacent cross-coupled segments are configured to conduct electrical current in opposing rotational directions.
 10. The transmitter of claim 8, the RF power source comprising: a plurality of phase-locked amplifiers disposed in at least a portion of the plurality of cross-coupled segments.
 11. The transmitter of claim 8, wherein the plurality of cross-coupled segments are disposed about the predetermined wireless power transfer area according to a pattern of endless knot wherein adjacent sides of the adjacent cross-coupled segments are electrically connected together in series.
 12. The transmitter of claim 8, wherein the pre-determined wireless power transfer area comprises a dimension exceeding a wavelength corresponding to the characteristic frequency of the variable form factor transmitter.
 13. The transmitter of claim 8, wherein: the respective transmitter loop for each of the plurality of cross-coupled segments comprises a pair of terminals electrically connected to a pair of terminals of an adjacent one of the plurality of cross-coupled segments; and electrical current flowing through the respective transmitter loop for each of the plurality of cross-coupled segments induces a magnetic field substantially orthogonal to a loop area of the respective transmitter loop.
 14. A system for wireless power transfer, comprising: a variable form factor transmitter having a form factor changeable by a user, adapted to form a plurality of cross-coupled segments, each forming a respective transmitter loop, disposed about a pre-determined wireless power transfer area, the respective transmitter loops formed from a string of distributed capacitors having a plurality of capacitors connected in series with one another via a plurality of wire segments, wherein each of the plurality of capacitors comprises a pre-determined capacitance, and each of the plurality of wire segments comprises a pre-determined segment length and a pre-determined inductance per unit length; and a radio frequency (RF) power source coupled to the plurality of cross-coupled segments, adjacent ones of the plurality of cross-coupled segments inducing respective magnetic fields having opposite direction to each other when driven together by the RF power source, wherein the plurality of cross-coupled segments are configured to: transmit, from the RF power source and based at least in part on a characteristic frequency dependent on at least the pre-determined capacitance and the pre-determined inductance per unit length, RF power across the pre-determined wireless power transfer area via a near electromagnetic field of the plurality of cross-coupled segments; and reduce, based on opposing directions of magnetic fields induced by adjacent cross-coupled segments of the plurality of cross-coupled segments, a radiation loss of the wireless power transfer due to a far electromagnetic field of the plurality of cross-coupled segments.
 15. The system of claim 14, wherein each of the plurality of cross-coupled segments comprises a plurality of sides; and wherein adjacent sides of the adjacent cross-coupled segments are configured to conduct electrical current in opposing rotational directions.
 16. The system of claim 14, the RF power source comprising: a plurality of phase-locked amplifiers disposed in at least a portion of the plurality of cross-coupled segments.
 17. The system of claim 14, wherein the plurality of cross-coupled segments are disposed about the predetermined wireless power transfer area according to a pattern of endless knot wherein adjacent sides of the adjacent cross-coupled segments are electrically connected together in series.
 18. The system of claim 14, wherein the pre-determined wireless power transfer area comprises a dimension exceeding a wavelength corresponding to a characteristic frequency of the plurality of cross-coupled segments.
 19. The system of claim 14, further comprising: a plurality of receiver devices disposed within the pre-determined wireless power transfer area; and wherein a portion of the RF power transmitted via the plurality of cross-coupled segments from the RF power source is received by the plurality of receiver devices.
 20. The system of claim 14, wherein: the respective transmitter loop for each of the plurality of cross-coupled segments comprises a pair of terminals electrically connected to a pair of terminals of an adjacent one of the plurality of cross-coupled segments; and electrical current flowing through the respective transmitter loop for each of the plurality of cross-coupled segments induces a magnetic field substantially orthogonal to a loop area of the respective transmitter loop. 